The לוח דבר בעיתו explains (regarding [Rosh Chodesh Iyar 5769][1]) that the reason we don't announce בה"ב on Rosh Chodesh is that:

>  אין מברכים את המתענים **בה"ב** בחדש ניסן או בשבת ר"ח, השבת הקרובה, כמו שנאמר ([קהלת ז, יד][2]) "**בְּיוֹם טוֹבָה - הֱיֵה בְטוֹב**", ומברכים בשבת שלאחריה.‏

I.e. *In a time of good fortune, enjoy the good fortune* ([Koheles 7:14][2]) - so we don't want to mention the fast (which is to atone for the *sins* caused by the *good times*) during those days which are happy - i.e. the month of Nissan and the demi-festival of Rosh Chodesh.

As in: Why dampen the atmosphere by mentioning an upcoming fast.

His source seems to be the [MaHaril in his chapter on סדר שני וחמישי][3]. (Hat tip: [Joel K][4])


  [1]: http://www.hebrewbooks.org/pdfpager.aspx?req=48214&st=&pgnum=906&hilite=
  [2]: https://www.sefaria.org.il/Ecclesiastes.7.14?lang=bi&with=all&lang2=he
  [3]: http://hebrewbooks.org/pdfpager.aspx?req=42570&st=&pgnum=49
  [4]: https://judaism.stackexchange.com/users/14850/joel-k